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Full text of "Treatise On Analysis Vol-Ii"

9    METRIZABLE GROUPS       39

9. METRIZABLE GROUPS

If G is a group, a function / on G x G is said to be left (resp. right)
invariant if f(xy, xz) =/0, z) (resp. f(yx, zx) =f(y, z)) for all x, y, z in G.
When G is commutative, these two conditions coincide, and/is then said to be
translation-invariant. A distance d on G is left (resp. right) invariant if and only
if the left (resp. right) translations are isometrics with respect to d. If / is a
left-invariant function on G x G, then the function (x, y')^-^f(x"19 y"1) is
right-invariant, and vice versa.

For example, if E is a normed space, the distance ||jc  >>|| on E is transla-
tion-invariant.

(1 2.9.1 ) In order that the topology of a topological group G should be metriz-
able (in which case G is said to be a metrizable group) it is necessary and
sufficient that there should exist a denumerable fundamental system of neigh-
borhoods of the neutral element #, whose intersection consists of e alone.
When this condition is satisfied, the topology of G can be defined by a left-
invariant distance , or by a right-invariant distance.

It is enough to show that if there exists a denumerable fundamental system
of neighborhoods (Un) of e in G such that Q Un = {e}, then the topology of G

n

can be defined by a left-invariant distance. We define inductively a sequence
(Vn) of symmetric neighborhoods of e in G such that Vj c: V1 and

for all n k 1 (12.8.3), so that (Vn) is also a fundamental system of neigh-
borhoods of e. Now define a real-valued function g on G x G as follows :
g(x, x) = 0; if x * y, then either x" 1y $ Vx, in which case we take g(x, y) = 1 ;
or else there exists a greatest integer k such that x~1y e Vk (because x~ly ^ e
cannot belong to all the Vn), in which case we define g(x, y) = 2"~k. It is
clear from this definition that g(x, y) = g(y, x), that g(x, y) ;> 0, and that
g(zx, zy) = g(x, y) for all x, y, z in G.
Now let

(12.9.1.1)                       i(x,y)=in

where the infimum is taken over the set of all finite sequences (z0 , zi9 . . . , zp)
(with p arbitrary) such that ZQ = x and zp = y. We shall show that d is a left-
invariant distance on G and satisfies the inequalities

(12.9.12)                     tg(x, y) ^ d(x9 y) ^ g(x, y).roup so defined is called the product of the topological groups G ,